$\dfrac{d}{dx}\left(x^3\sin(x)\right)=$
$x^3\sin(x)$ is the product of two, more basic, expressions: $x^3$ and $\sin(x)$. Therefore, the derivative of the expression can be found using the product rule : $\begin{aligned} \dfrac{d}{dx}[u(x)v(x)]&=\dfrac{d}{dx}[u(x)]v(x)+u(x)\dfrac{d}{dx}[v(x)] \\\\ &=u'(x)v(x)+u(x)v'(x) \end{aligned}$ $\begin{aligned} &\phantom{=}\dfrac{d}{dx}\left(x^3\sin(x)\right) \\\\ &=\dfrac{d}{dx}\left(x^3\right)\sin(x)+x^3\dfrac{d}{dx}(\sin(x))&&\gray{\text{The product rule}} \\\\ &=3x^2\cdot\sin(x)+x^3\cdot \cos(x)&&\gray{\text{Differentiate }x^3\text{ and }\sin(x)} \\\\ &=x^2(3\sin(x)+x\cos(x))&&\gray{\text{Simplify}} \end{aligned}$ In conclusion, $\dfrac{d}{dx}\left(x^3\sin(x)\right)=x^2(3\sin(x)+x\cos(x))$ or any other equivalent form.